Tuesday, March 28, 2017

This post calls attention to A Little History of Economics, by Niall Kishtainy. Since I only skimmed this in a local bookstore, this post is not a proper review.

Kishtainy's book, in successive chapters, focuses on the lives of specific economists. The concluding chapter
asks why one would want to be an economist, a question that is answered, I gather, by the preceding chapters.
The book is in the
same genre as Robert Heilbroner's The Worldly Philosophers. Since many more economists are covered in a short span,
individual chapters are shorter. As I recall, economists (not in this order) covered include:

Augustine and Thomas Aquinas

Mercantilists

Francois Quesnay, Mirabeau, and other Physiocrats

Adam Smith

David Ricardo

Charles Fourier, Robert Owen, and other utopian socialists

Thomas Malthus

Friedrich List

Karl Marx

William Stanley and Alfred Marshall

German historical school, Austrian school, and the methodenstreit

Thorstein Veblen

Vladimir Lenin and John Hobson

Ludwig von Mises

Joan Robinson and Edward Chamberlin

John Maynard Keynes

Paul Samuelson, J. R. Hicks

Friedrich Hayek

Arthur Lewis, Paul Rosenstein-Rodan, and Raul Prebisch

Robert Solow, Trevor Swan, and Paul Romer

Joseph Schumpeter

Gary Becker

John Von Neumann and John Nash

Ken Arrow and Gerard Debreu

Fidel Castro, Che Guevara, and Andre Frank

James Buchanan

Milton Friedman

George Akerlof and Joseph Stiglitz

Hyman Minsky

John Muth, Eugene Fama, and Robert Lucas

Ed Prescott and Finn Kyland

Behavioral economics

Thomas Piketty

Doubtless, if I read the book in detail, I would have objections to specific details. The attempted coverage, however,
seems quite impressive. This book extends from before to after most histories. And it covers a wider range of
economists than most.

Wednesday, March 22, 2017

Over on EconSpeak, Bruce Wilder provides
somecomments on a post. He notes
that economists wanting to criticize glib free-market ideology in the public
discourse often seem unwilling to discard neoclassical economic theory.

Paul Krugman illustrates how theoretically conservative and neoclassical
a liberal economist can be.
(I use "liberal" in the sense of contemporary politics in the USA.)
I refer to Krugman's post
from earlier this week,
in which he adapts an analysis from the theory of international trade to consider
technological innovation (e.g., robots). Krugman presents a diagram, in which endowments of capital and labor
are measured along the two axes. Krugman does not seem to be aware that one cannot, in general, coherently
talk about a quantity of capital, prior to and independently of prices.
He goes on to talk about "capital-intensive" and "labor-intensive" techniques of production.

I point to my draft paper,
"On the loss from trade", to illustrate my point that one cannot meaningfully talk about the
endowment of capital.

(I did submit this paper to a journal. A reviewer said it was not original enough.
I emphasized that I was illustrating my points
in a flow-input, point output model, with a one-way flow from factors of production
to consumption goods, not a model of production of commodities by means of commodities.
Steedman & Metcalfe (1979) also has a one-way model, albeit with a point-input, point-output
model. So the reviewer's comments were fair. Embarrassingly, I
cite other papers from that book. Apparently, I had forgotten that paper, if I ever
read it. I suppose that, given the chance, I could have distinguished some of my points from
those made in Steedman & Metcalfe (1979). Also, I close my model with utility-maximization;
if I recall correctly, Steedman leaves such an exercise to the reader in papers in
that book.)

Saturday, March 18, 2017

Figure 1: Rates of Profits for Switch Points for Differential Rates of Profits

1.0 Introduction

Suppose one knows the technology available to firms at a given point in time. That is, one knows the techniques
among which managers of firms choose. And suppose one finds that reswitching cannot occur under this
technology, given prices of production in which the same rate of profits prevails among all industries.
But, perhaps, barriers to entry persist. If one analyzes the choice of technique for the
given technology, under the assumption that prices of production reflect stable (non-unit) ratios
of profits, differing among industries, reswitching may arise for the technology. The numerical
example in this post demonstrates this logical possibility.

The numerical example follows a model of oligopoly I have previously outlined. In some sense, the example is symmetrical to the example in this
draft paper.
That example is of a reswitching example under pure competition, which becomes an example without reswitching
and capital reversing, if the ratio of the rates of profits among industries differs enough.
The example in this post, on the other hand, has no reswitching or capital reversing under pure
competition. But if the ratios of the rates of profits becomes extreme enough, it becomes a reswitching
example.

2.0 Technology

The technology for this example resembles many I have explained in past posts.
Suppose two commodities, iron and corn, are produced in the example economy.
As shown
in Table 1, two processes are known for producing iron, and one corn-producing process is known.
Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column.
All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital.
Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes.

Table 1: The Technology for a Two-Industry Model

Input

IronIndustry

CornIndustry

Labor

1

305/494

1

Iron

1/10

229/494

11/10

Corn

1/40

3/1976

2/5

For the economy to be self-reproducing, both iron and corn must be produced each year. Two
techniques of production are available. The Alpha technique consists of the first iron-producing
process and the lone corn-producing process. The Beta technique consists of the remaining
iron-producing process and the corn-producing process.

3.0 Price Equations

The choice of technique is analyzed on the basis of cost-minimization, with prices of production.
Suppose the Alpha technique is cost minimizing. Then the following system of equalities and
inequalities hold:

[(1/10)p + (1/40)](1 + rs1) + w = p

[(229/494)p + (3/1976)](1 + rs1) + (305/494)w ≥ p

[(11/10)p + (2/5)](1 + rs2) + w = 1

where p is the price of a unit of iron, and w is the wage.

The parameters
s1 and s2 are given constants, such that rs1
is the rate of profits in iron production and rs2 is the rate of profits
in corn production. The quotient s1/s2 is the ratio,
in this model, of the rate of profits in iron production to the rate of profits in
corn production. Consider the special case:

s1 = s2 = 1

This is the case of free competition, with investors having no preference among industries.
In this case, r is the rate of profits. I call r the scale factor for the
rate of profits in the general case where s1 and s2
are unequal.

The above system of equations and inequalities embody the assumption that a unit corn
is the numeraire. They also show labor as being advanced and wages as paid out of
the surplus at the end of the period of production. If the second inequality is
an equality, both the Alpha and the Beta techniques are cost-minimizing; this is
a switch point. The Alpha technique is the unique cost-minimizing technique if it
is a strict inequality. To create a system expressing that the Beta technique
is cost-minimizing,
the equality and inequality for iron production are
interchanged.

4.0 Choice of Technique

The above system can be solved, given s1, s2,
and the scale factor for the rate of profits. I record the solution for a couple
of special cases, for completeness. Graphs of wage curves and a bifurcation
diagram illustrate that stable (non-unitary) ratios of rates of profits can
change the dynamics of markets.

4.1 Free Competition

Consider the special case of free competition.
The wage curve for the Alpha technique is:

wα = (41 - 38r + r2)/[80(2 + r)]

The price of iron, when the Alpha technique is cost-minimizing, is:

pα = (5 - 3r)/[8(2 + r)]

The wage curve for the Beta technique is:

wβ = (6,327 - 9,802r + 3,631r2)/[20(1,201 + 213r)]

When the Beta technique is cost-minimizing, the price of iron is:

pβ = [5(147 - 97r)]/[2(1,201 + 213r)]

Figure 2 graphs the wage curves for the two techniques, under free competition
and a uniform rate of profits among industries.
The wage curves intersect at a single switch point, at
a rate of profits of, approximately, 8.4%:

rswitch = (1/1,301)[799 - 24 (8261/2)]

The wage curve for the Beta technique is on the outer envelope, of the wage curves,
for rates of profits below the switch point. Thus, the Beta technique
is cost-minimizing for low rates of profits. The Alpha technique is cost minimizing
for feasible rates of profits above the switch point. Around the switch point,
a higher rate of profits is associated with the adoption of a less capital-intensive
technique. Under free competition, this is not a case of capital-reversing.

Figure 2: Wage Curves for Free Competition

4.2 A Case of Oligopoly

Now, I want to consider a case of oligopoly, in which firms in different
industries are able to ensure long-lasting barriers to entry. These
barriers manifest themselves with the following parameter values:

s1 = 4/5

s2 = 5/4

In this case, the wage curve for the Alpha technique is:

wα = (4,100 - 4,435r + 100r2)/[40(400 + 259r)]

The price of iron, when the Alpha technique is cost-minimizing, is:

pα = (125 - 96r)/(400 + 259r)

The wage curve for the Beta technique is:

wβ = 8(126,540 - 195,289r + 72,620r2)/[160(24,020 + 9,447r)]

The price of iron, when the Beta technique is cost-minimizing, is:

pβ = 2(3,675 - 3,038r)/(24,020 + 9,447r)

Figure 3 graphs the wage curves for the Alpha and Beta techniques, for the
parameter values for this model of oligopoly. This is now an example of
reswitching. The Beta technique is cost minimizing at low and high rates
of profits. The Alpha technique is cost minimizing at intermediate rates.
The switch points are at, approximately, a value of the scale factor for
rates of profits of 12.07% and 77.66%, respectively.

Figure 3: Wage Curves for a Case of Oligopoly

4.3 A Range of Ratios of Profit Rates

The above example of oligopoly can be generalized. I restrict myself to the case where the parameters expressing
the ratio of rates of profits between industries satisfy:

s2 = 1/s1

One can then consider how the shapes and locations of wage curves and switch points vary with continuous
variation in s1/s2. Figure 1, at the top of this post, graphs the
wage at switch points for a range of ratios of rates of profits. Since the Beta technique is cost-minimizing,
in the graph, at all high feasible wages and low scale factor for the rates of profits, I only graph the
maximum wage for the Beta technique. I do not graph the maximum wage for the Alpha technique.

As the ratio of the rate of profits in the iron industry to rate in the corn industry increases towards unity,
the model changes from a region in which the Beta technique is dominant to a reswitching example to an
example with only a single switch point. As expected, only one switch point exists when the rate of
profits is uniform between industries.

5.0 Conclusion

So I have created and worked through an example where:

No reswitching or capital-reversing exists under pure competition, with all industries earning the same rate of profits.

Reswitching and capital-reversing can arise for oligopoly, with persistent differential rates of profits across industries.

No qualitative difference necessarily exists, in the long period theory of prices,
between free competition and imperfections of competition.
Doubtless, all sorts of complications of strategic behavior, asymmetric information, and so on
are empirically important. But it seems confused to blame the failure of markets to clear
or economic instability on such imperfections.

Wednesday, March 15, 2017

Figure 1: Rates of Profits for Switch Points in One Dimension in Parameter Space

1.0 Introduction

This post presents an example of structural variation in the qualitative behavior of a reswitching
example, at different values for selected parameters. I know of few applications of bifurcation
analysis to the Cambridge Capital Controversy. Most prominently, I think of Rosser (1983). I suppose
I could also point to some of my draftpapers. Although not presented this
way, one could read Laibman and Nell (1977) as a bifurcation analysis, where the steady state rate of growth is
the parameter being varied.

I guess one could read this post as a response to the empirical results in Han and Schefold (2006).
Schefold has been developing a theoretical explanation, based on random matrices, of why capital-theoretic
paradoxes might be empirically rare. I seem to have stumbled on an explanation of why such paradoxes might
arise in practice, and yet might not be observable without more data. To fully address recent results
from Schefold, on reswitching and random matrices, one should analyze the spectra of Leontief
input-output matrices, which I do not do here.

2.0 Technology

Suppose two commodities, iron and corn, are produced in the economy in the numerical example. As shown
in Table 1, two processes are known for producing iron, and one corn-producing process is known.
Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column.
All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital.
Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes.

Table 1: The Technology for a Two-Industry Model

Input

IronIndustry

CornIndustry

Labor

1

305/494

a0,2

Iron

1/10

229/494

a1,2

Corn

1/40

3/1976

2/5

Assume a0,2 is non-negative, and that a1,2 is strictly positive.
For the economy to be self-reproducing, both iron and corn must be produced each year. Two
techniques of production are available. The Alpha technique consists of the first iron-producing
process and the lone corn-producing process. The Beta technique consists of the remaining
iron-producing process and the corn-producing process.

3.0 Choice of Technique

Managers of firms choose the technique to adopt based on cost-minimization. I take a bushel
of corn as the numeraire. Assume that labor is advanced, and that wages are paid out of
the surplus at the end of the year. For this post, I do not bother setting out equations
for prices of production; I have done that many times in the past.

3.1 Reswitching for one Set of Parameter Values

Figure 2 illustrates that this is a reswitching example. This figure is drawn for
the following values of the labor coefficient in the process for producing corn:

a0,2 = 1

The coefficient of production for iron in corn-production, in drawing Figure 2, is set to
the following value:

a1,2 = 2

The economy exhibits capital-reversing around the switch point at an 80% rate of profits.

Figure 2: Wage-Rate of Profits Curves

3.2 Bifurcations with Variations in a Labor Coefficient

Wage-rate of profits curves are drawn for given coefficients of production. And they will be moved
elsewhere for different levels of coefficients of production. Consequently, the existence and
location of switch points differ, depending on the values for coefficients of production.

Accordingly, suppose all coefficients of production, except a0,2, are as in the
above reswitching example. Consider values of the labor coefficient for corn-production ranging
from zero to three. The labor coefficient is plotted along the abscissa in Figure 3. The points
on the blue locus in the figure show the rate of profits for the switch points, as a correspondence
for the labor coefficient. The maximum rates of profits for the Alpha and Beta techniques are
also graphed.

Figure 3: Rates of Profits for Switch Points as One Labor Coefficient Decreases

Figure 3 shows a structural change in the example. Up to a value of a0,2 of approximately
2.74, this is a reswitching example. For parameter values strictly greater than that, no switch points
exist. The maximum rates of profits for the two techniques are constant in Figure 3.
The maximum rates of profits are found for a wage of zero, and they do not vary with the labor
coefficient. In some sense, only the maximum rate of profits for the Beta technique is relevant
in the figure.

3.3 Bifurcations with Variations in a Coefficient of Production for Iron

Figure 1, at the top of this post, also shows structural changes. The coefficient of
production for iron in corn-production varies in the figure. a1,2
ranges from one to three. The other coefficients of production are as in the reswitching
example in Section 3.1 above. And the blue locus shows the rate of profits at switch
points.

The example can seen to have structural variations here, also, with three distinct
regions for a1,2, with the same qualitative behavior in each
region. For a low enough value of the coefficient of production under consideration,
only one switch point exists. The model remains a reswitching example for an
intermediate range of this parameter. And for values of this coefficient
of production strictly greater than approximately 2.53,
the Beta technique is cost-minimizing for all feasible wages and rates of profits.

The maximum rates of profits, for the Alpha and Beta techniques, are also graphed
in Figure 1.

4.0 A Story of Technological Process

Using the above example, one can tell a story of technological progress. Suppose at the start of the story, corn production
requires a relatively large input of direct labor and iron, per (gross) unit corn produced. Prices of production
associated with this technology are such that only one technique is cost-minimizing. For all feasible wages
and rates of profits, firms will want to adopt the Beta technique.

Suppose iron production is relatively stagnant, as compared to corn-production. Innovation in the corn industry
reduces the labor and iron coefficients defining the single dominant corn-producing process. After some time,
either or both coefficients will be reduced enough that the technology for this economy will have become
a reswitching example. And around the switch point at the lower wage (and higher rate of profits), a higher
wage is associated with the cost-minimizing technique requiring more labor to be hired, in the overall economy,
per given bushel of corn produced (net).

But technological innovation continues to proceed apace.
At a even lower coefficient of production for the iron input in the corn industry, the structural behavior
of the economy changes again. Now a single switch point exists. And the results of the choice of
technique around that switch point conforms to outdated neoclassical intuition.

5.0 Conclusion

This example has two properties that I think worth emphasizing.

The choice of technique in the example corresponds to a choice of a production process in the
iron industry. As I have told the story, the technology is fixed in iron production. Innovation
occurs in corn production. Thus, innovation in one industry can change the dynamics in another
industry.

Second, suppose the technology is observed at a single point of time. Suppose the economy is more or less
stationary, and that observation is taken at either the start or the end of the above story. Then
neither reswitching nor capital reversing will be observed. Yet such phenomena might arise in the
future or have arisen in the past.

Saturday, March 11, 2017

A commentator informs me that the True Levelers revived some ideas put forth in the Peasants Revolt.

Another commentator points me to Naoki Yoshihara's review of Opocher and Steedman's recent book. Yoshihara has a point, but I think the practice of treating inputs and physically identical outputs as different dated commodities is less applicable in partial models, as opposed to full General Equilibrium. Accountants need guidelines that resist easy manipulation in calculating profits and losses.

Antonella Palumbo has a post, "Can 'It' Happen Again? Defining the Battlefield for a Theoretical Revolution in Economics", at the Institute for New Economic Thinking. Palumbo argues that a revival of classical economics, without Say's law, can provide an alternative to neoclassical economics. And Keynes' macroeconomics can be usefully be combined with this revival.

Wednesday, March 08, 2017

I think I may have an original criticism of (a good part of) neoclassical economics. For purposes of this post, I here define the use of continuously differential production functions as an essential element in the neoclassical theory of production. (This is a more restrictive characterization than I usually employ.) Consider this two-sector example, in which coefficients of production in both sectors varies continuously along the wage-rate of profits frontier. It would follow from this post, I guess, that neoclassical theory is a limit, in some sense, of an analysis in which all switch points are flukes.

I have presented many other, often unoriginal, examples with a continuum of techniques:

I have an example with an uncountably infinite number of techniques along the wage-rate of frontier, but discontinuities for (all?) marginal relationships.

2.0 Technology

I want to compare and contrast two models. The technology in the second model is an example in Salvadori and Steedman (1988).

Households consume a single commodity, called "corn", in both models. In both models, two
processes are known for producing corn. And these processes require inputs of labor and
a capital good to produce corn. All processes exhibit Constant Returns to Scale, and
all processes require services of inputs over a year to produce output of a single
commodity available at the end of the year. Both models are models of circulating
capital. Nothing remains at the end of the year of the capital goods whose services
are used by firms during the production processes.

2.1 First Model

The technology for the first model is shown in Table 1.
Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry
for that column.
The two processes for
producing corn require inputs of distinct capital goods. One corn-producing process
requires inputs of labor and iron, and the other requires inputs of labor and tin.

Table 1: The Technology for a Three-Industry Model

Input

IronIndustry

TinIndustry

CornIndustry

(a)

(b)

(c)

(d)

Labor

1

2

1

2

Iron

0

0

2/3

0

Tin

0

0

0

1/2

Corn

2/3

1/2

0

0

Two techniques, as shown in Table 2, are available for producing a net output of corn. A choice of a process for producing corn
also entails a choice of which capital good is produced. When the processes are each operated on a appropriate scale, the gross
output of the process producing the specific capital good exactly replaces the quantity of the capital good used up as an
input, summed over both industries operated in the technique.

Table 2: Techniques in a Three-Commodity Model

Technique

Processes

Alpha

a, c

Beta

b, d

2.2 Second Model

The technology for the second model is shown in Table 3. Two processes are known for producing corn. Both corn-producing
processes require inputs of labor and iron, but in different proportions.

Table 3: The Technology for a Two-Industry Model

Input

IronIndustry

CornIndustry

(a)

(b)

(c)

(d)

Labor

1

2

1

2

Iron

0

0

2/3

1/2

Corn

2/3

1/2

0

0

Table 4 lists the techniques available in the second model. The first two techniques superficially resemble the two
techniques available in the first model. But, in this model, the first process for producing a capital good can
be combined, in a technique, with the second corn-producing producing process. This combination of processes is
called the Gamma technique. Likewise, the Delta technique combines the second process for producing a capital
good with the first corn-producing processes. Nothing like the Gamma and Delta techniques are available in the
first model.

Table 4: Techniques in a Two-Commodity Model

Technique

Processes

Alpha

a, c

Beta

b, d

Gamma

a, d

Delta

b, c

3.0 Prices of Production

Suppose the Alpha technique is cost-minimizing. Prices of production, which permit smooth
reproduction of the economy, must satisfy the following system of two equations in three unknowns:

(2/3)(1 + r) + wα = pα

(2/3) pα(1 + r) + wα = 1

These equations are based on the assumption that labor is advanced, and wages are paid out of the surplus
at the end of the year. The same rate of profits are generated in both industries. A unit quantity of corn
is taken as the numeraire.

One of the variables in these equations can be taken as exogenous. The first row in Table 5 specifies the wage
and the price of the appropriate capital good, as a function of the rate of profits. The equation in the
second column is called the wage-rate of profits curve, also known as the wage curve,
for the Alpha technique. Table 5 also shows solutions of the systems of equations for the prices of production
for the other three techniques in the second model, above. I have deliberately chosen a notation such
that the first two rows can be read as applying to either one of the two models.

Table 5: Wages and Prices by Technique

Technique

Wage Curve

Prices

Alpha

wα = (1 - 2 r)/3

pα = 1

Beta

wβ = (1 - r)/4

pβ = 1

Gamma

wγ = 2(2 - 2r - r2)/[3(5 + r)]

pγ = 2(7 + 4r)/[3(5 + r)]

Delta

wδ = (2 - 2r - r2)/(7 + 4r)

pδ = 3(5 + r)/[2(7 + 4r)]

Figure 1, at the top of this post, graphs all four wage-curves. The wage curves for the Alpha and Beta techniques are
straight lines. In the jargon, the processes comprising these techniques exhibit the same organic composition of capital.
The wage curves for the Gamma and Delta techniques are not straight lines. All four wage-curves intersect at a single
point, (r, w) = (20%, 1/5). (The wage curves for the Gamma and Delta techniques have the same intersection
with the axis for the rate of profits.)

3.0 Choice of Technique

The cost-minimizing techniques form the outer envelope of the wage curves. For a given wage,
the cost minimizing technique is the technique with the highest wage curve in Figure 1.
A switch point is a point on the outer envelope at which more than one technique
is cost-minimizing. All four wage curves intersect, in the figure, at the single switch point.

The Beta technique is cost-minimizing for wages to the left of the single switch point.
The Alpha technique is cost-minimizing
for all feasible wages greater than the wage at the switch point. Managers of firms
replace both processes in the Alpha technique at the switch point with both processes
in the Beta technique.

This is no problem for the first model above. The adoption of a new process for producing
corn requires, if the economy is capable of self-replacement before and after the switch,
that the process for producing iron or tin be replaced by the process for producing the
other.

But consider the other model. For all processes in the Alpha technique to be replaced at a
switch point, the wage curves for all techniques composed of all combinations of processes
in the Alpha and Beta techniques. In other words, in the second model, wage curves
for all four techniques must intersect at the switch point. The example in the
second model is a fluke.

I have previously
explained
what makes a result a fluke, in the context of the analysis of the choice of technique.
Qualitative properties, for generic results, continue to persist for some small variation in model parameters.

Consider a model with a discrete number of switch points.
Consider the cost-minimizing techniques on both sides of a switch point. And
suppose that same commodities are produced in both techniques, albeit in different
proportions. Generically, only one process is replaced at such a switch point.
All processes, except for that one, are common in both techniques.

5.0 A Generalization to An Uncountably Infinite Number of Processes in Each Industry

Consider a model with more than one industry, but a finite number.
Suppose each industry has available an uncountably infinite number of processes. And,
in each industry, the processes available for that industry can be described by a
continuously differentiable production function.
Here
I present a two-commodity example with Cobb-Douglas production functions.

There are no switch points in such a model. The cost-minimizing technique varies
continuously along the outer-envelope of wage curves. In fact, the processes
in each industry, in the cost-minimizing technique varies continuously.
Since there are no switch
points at all, there is not a single switch point in which more than one process varies,
as a fluke, with the cost-minimizing technique.

Nevertheless, cannot one see such "smooth" production functions as a limiting case?
If so, it would be a generalization or extension of a discrete model, in which all switch
points are flukes, to a continuum. From the perspective of the analysis of
the choice of technique in discrete models, typical neoclassical models
are nothing but flukes.

6.0 Conclusions

I actually found my negative conclusion surprising. I have tried to be conscious of the distinction
between the structure of the two models in Section 2 above. I think at least some
examples I have presented cannot be attacked by the above critique.
They are examples of the first, not the second model. I tend to read Samuelson (1962)
in the same way, as not sensitive to the critique in this post.

Saturday, March 04, 2017

I have previously considered all roots of a polynomial equation for the rate of profits in a model, of the choice of technique, in which each technique is specified by a finite series of dated labor inputs. One root is the traditional rate of profits, but there are uses for the other roots:

All roots appear in an equation defining the Net Present Value (NPV) for the technique, given the wage and the rate of profits.

All roots can be combined in an accounting identity for the difference between labor commanded and labor embodied, given the wage.

I thought it of interest to know whether these non-traditional roots are real or complex, as they vary with the wage. I am considering multiple roots in an attempt to build on and critique Michael Osborne's approach to multiple interest rate analysis.

I also have considered examples of models of the production of commodities by means of commodities, in which at least one commodity is basic, in the sense of Sraffa. And I have attempted to apply or extend my critique of multiple interest rate analysis to these models.
The point of this post is to illustrate possibilities on the complex plane for multiple interest rates in these models.

A technique in models of the production of commodities by means of commodities, as least in the case when all capital is circulating capital,
is specified as a vector of labor coefficients and a Leontief input-output matrix.
In parallel with my approach to techniques specified by a finite sequence of dated labor inputs, consider wages as being advanced - that is, not paid at the end of the year out of the surplus - in such models. Given the wage and the numeraire, one can construct a square matrix in which each coefficient
is the sum of the corresponding coefficient in the Leontief input-output matrix and the quantity of the commodity produced by that industry that is advanced to the workers, per unit output produced. I call this matrix the augmented input-output matrix.

A polynomial equation, called the characteristic equation, is solved to find eigenvalues of the augmented
input-output matrix. The power of this polynomial is equal to the number of commodities produced by the technique.
The number of roots for the polynomial is therefore equal to the number of commodities.
A rate of profits corresponds to each root.
Assume the Leontief input-output matrix is a Sraffa matrix and that the wage does not
exceed a certain maximum.
Under these conditions, the Perron-Frobenius theorem picks out the maximum eigenvalue of the
augmented input-output matrix. The corresponding rate of profits is non-negative, and
the prices of production of these commodities are positive at the given wage.
I was not able to find an application for the other, non-traditional rates of profits.

I present a numerical example in this working paper.
This is a three-commodity example with two techniques. Figure 1 graphs the three roots, at different level of wages, for the Beta technique
in that example.

In a previous blog post, I
extend that example such that managers of firms have a choice of process for producing each of the three commodities.
As a consequence, a choice among eight techniques arises. And one can draw a graph like Figure 1 for each technique
in that example. Figure 2 shows the corresponding graph for the Delta technique.

Figure 2: Rates of Profits for Delta Technique

In Figures 1 and 2, the rate of profits
picked out by the Perron-Frobenius theorem and used to draw the wage-rate of profits curve for the technique lies along the line segment on the real
axis on the left in the figure. A lower wage corresponds to a higher traditional rate of profits. Thus, points further to the right
on this line segment correspond to a lower wage. A wage of zero leads to the right-most point on this line segment. The highest
feasible wage corresponds to left-most point, at a rate of profits of zero, on this segment.

Two non-traditional rates of profits arise for the other two solutions of the characteristic equation.
They are plotted to the right on the graphs in Figures 1 and 2.
When complex, they are complex conjugates.
I thought it of interest that, in Figure 2, they are purely real for two non-overlapping, distinct ranges of feasible levels of
the wage.

I draw no practical, applied implications from the non-traditional rates of profits. I just think the graphs are curious.

Thursday, March 02, 2017

Bill Black has one here, emphasizing Arrow's impossibility theorem. The blog, A Fine Theorem, has two of a planned four-part series. The first is on the impossibility theorem. The second is about General Equilibrium. The two planned, I gather, are to be about learning-by-doing and health economics, respectively.

I have written several posts drawing on Arrow's work. This one, on a sophisticated neoclassical response to the Cambridge Capital Controversy, is among my most popular posts.